課程名稱：電磁學上 課程性質：物理系必帶 課程教師：王立民 開課學院：理學院 開課系所：物理系 考試日期（年/月/日）：99/10/21 考試時限（分鐘）：120分鐘（8:10~10:10） 是否需發放獎勵金：是 範圍：Introduction to Electrodynamics 3rd edition CH1~CH2.4 1.Assume a vector function v=(x^2*y)i + (-x*y^2)j. Test Stokes' theorem: (a)Find ∮v．dl around the triangular contour shown in Fig.1.(10%) (b)Evaluate ∫(▽×v)．da over the triangular area.(10%) ↑ │　　　　　．(1,2) │　　　　↗│ Fig.2　為一半徑r=1，球心(0,0,0) │　　　╱　│ 的半球面在z>0的部分 │　　╱　　│ │　╱　　　│ │╱　　　　↓ ┼←────┴───→ │(0,0) (1,0) Fig.1 2.A vector field D=[r(cosψ)^2] r(unit radius vector) exists in the region over the hemisphere with radius of r=1 as shown in Fig.2. For checking the divergence theorem, evaluate: (a)∮D．da (10%) (b)∫(▽．D)dτ (10%) 1 δ 1 δ 1 δ(Dψ) ▽．D=── ──(r^2*Dr)+──── ──(sinθ*Dθ)+──── ─── r^2 δr r*sinθ δθ r*sinθ δψ (δ:偏微分) 3.A spherical distribution of charge ρ=ρ0*(1-r/b) exist in the region 0<=r<=b, where ρ0 and b are constant. Determine: (a)The total charge Q.(5%) (b)Electric field Einside and Eoutside.(5%) (c)Electric potentials Vinside and Voutside.(5%) (d)Electrostatic energy W stored.(5%) 4.A uniformly charged sphere of radii R centered at the origin (point A) has a spherical cavity of radii R/2 centered at (x=R/2,y=0,z=0). The total charge of the solid part is Q. (a)Find the electric field E for any point in the cavity.(10%) (b)Calculate the potential difference between point A and B, where point B is at (x=R,y=0,z=0).(10%) 5.A long coaxial cable (Fig.4) carries a uniform nolume positive charge density ρ on the inner cylinder (radius a), and a uniform negative surface charge density on the outer cylindrical shell (radius b). The cable is electrically neutral. (a)Find the potential difference between a point on the axis and a point on the outer cylindrical shell.(10%) (b)Calculate the energy per unit length of this configuration.(10%) Fig.4 有一內半徑為a的帶正電實心圓柱，外面包著外半徑為b的帶負電圓柱形薄殼。 6.Consider two concentric spherical shells, of radii a and b. Suppose the inner one carries a charge 2q, and the outer one a charge -q (both uniformly distributed).Calculate the energy of this configuration in two different ways.(20%) --

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